Electronic Materials

# Crystal lattice and miller indices

This post is about structure of solid state materials, their properties and quantitative characteristics.

### Crystal lattice

A crystal lattice is the basis of the order in the environment of molecules and atoms. It provides periodic electrostatic fields across the compound. The periodicity of the structure is the key characteristic of crystals. Formation of a lattice by some elements or molecules, depends on their size and the electronic configuration of their outer orbits. Geometrically there are 14 lattices possible, and they are based on six crystal systems. Geometrical classification is not the only way to classify solids by their lattice. Solids also differ according to their refraction, magnetic features. There is 32 classes of symmetry, where 230 spatial groups are possible.

Here are all possible crystal lattices types:

Table 1.3.

 Crystal system Interaxial angles Axial relationships Cubic α = β = γ = 90 a = b = c Hexagonal α = β = 90, γ = 120 a = b ≠ c Tetragonal α = β = γ = 90 a = b ≠ c Rhombohedral α = β = γ ≠ 90 a = b = c Orthorhombic α = β = γ = 90 a ≠ b ≠ c Monoclinic α = γ = 90 ≠ β a ≠ b ≠ c

Crystal compounds can be in the form of one monocrystal, or consist of many small crystals (grains). In the case of polycrystal (with crystal grains) atoms in the structure have some periodicity in a grain. However, periodical structure breaks between some grains. So, on the borders between grains, there is no periodical structure between grains. Monocrystals are characterised by anisotropy. Anisotropy is the difference in properties of a solid in different directions. However, there are no anisotropy in polycrystals. Talking about the properties of monocrystals we have to mention their crystallography planes and directions. Miller indexes help us to describe these parameters.

### Miller indices

Crystallographic planes in all lattices are specified by Miller indices ($hkl$). Any planes parallel to each other are equivalent and have the same indices. This is how Miller indices can be determined:

1. A crystallographic plane in a solid should be translated to the considered solid compound unit cell.
2. A crystallographic plane intersects each of the three axis or parallel to some of them. The intersection point should be considered as in accordance to the axis.
3. If a plane is parallel to an axis, it is considered to have $0$ index.
4. Then reciprocal values have to be determined:
$\frac{a}{A},\frac{b}{B},\frac{c}{C}.$ If needed these numbers have to be changed to the smallest integers, by division of the common coefficient.
5. Finally, Miller indices are ($hkl$), and can be determined by the following equations:
Where n – is the common coefficient, reducing $h$$k$ and $l$ to integers.